Theorem imaeq2 5037

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4973	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4926	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4706	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4706	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2265	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1373   ran crn 4694   |` cres 4695   "cima 4696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  imaeq2i  5039  imaeq2d  5041  fimadmfo  5529  ssimaex  5663  ssimaexg  5664  isoselem  5912  f1opw2  6175  fopwdom  6958  ssenen  6973  fiintim  7054  fidcenumlemrk  7082  fidcenumlemr  7083  sbthlem2  7086  isbth  7095  ennnfonelemp1  12892  ennnfonelemnn0  12908  ctinfomlemom  12913  ctinfom  12914  tgcn  14795  iscnp4  14805  cnpnei  14806  cnima  14807  cnconst2  14820  cnrest2  14823  cnptoprest  14826  txcnp  14858  txcnmpt  14860  metcnp3  15098